Symmetry actions and brackets for adjoint-symmetries. I: Main results and applications

Abstract Infinitesimal symmetries of a partial differential equation (PDE) can be defined algebraically as the solutions linearisation (Frechet derivative) holding on space to PDE, and they are well-known comprise linear having structure Lie algebra. Solutions adjoint PDE called adjoint-symmetries. Their algebraic for general systems is studied herein. This motivated by correspondence between variational conservation laws arising from Noether’s theorem, which has modern generalisation non-variational PDEs, where infinitesimal replaced adjoint-symmetries, multipliers (adjoint-symmetries satisfying certain Euler-Lagrange condition). Several main results obtained. Symmetries shown have three different actions These used construct bilinear adjoint-symmetry brackets, one pull-back symmetry commutator bracket properties bracket. The brackets do not use or require existence any local (Hamiltonian Lagrangian) thus apply systems. One encode pre-symplectic (Noether) operator, leads construction symplectic 2-form Poisson evolution generalised KdV in potential form illustrate all results.